<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">Amp</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>, <font color="Maroon">Amp'</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:70"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="gcd_1.html#R5" target="_self">are_canonical_wrt</a> <font color="Maroon">Amp</font>
 ;<br/>

<a NAME="E2:70"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:70"/><b>then </b><i><font color="Green">E20</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:70"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E5:70"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E6:70"/><i><font color="Green">E26</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:70"/><i><font color="Green">E53</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E8:70"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:70"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">Amp'</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E7:70"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="gcd_1.html#T14" target="_self">Th14</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:70"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="gcd_1.html#R5" target="_self">are_canonical_wrt</a> <font color="Maroon">Amp'</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T10" target="_self">Th10</a></i>;<br/>


</div>
